{
  "id": "1901.00476",
  "version": "v1",
  "published": "2018-12-28T21:51:52.000Z",
  "updated": "2018-12-28T21:51:52.000Z",
  "title": "Further Combinatorial Identities deriving from the $n$-th power of a $2 \\times 2$ matrix",
  "authors": [
    "James Mc Laughlin",
    "Nancy J. Wyshinski"
  ],
  "comment": "9 pages",
  "journal": "Discrete Applied Mathematics 154 (2006), no. 8, 1301--1308",
  "doi": "10.1016/j.dam.2006.01.003",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we use a formula for the $n$-th power of a $2\\times2$ matrix $A$ (in terms of the entries in $A$) to derive various combinatorial identities. Three examples of our results follow. 1) We show that if $m$ and $n$ are positive integers and $s \\in \\{0,1,2,\\dots,$ $\\lfloor (mn-1)/2 \\rfloor \\}$, then \\begin{multline*} \\sum_{i,j,k,t}2^{1+2t-mn+n} \\frac{(-1)^{nk+i(n+1)}}{1+\\delta_{(m-1)/2,\\,i+k}} \\binom{m-1-i}{i} \\binom{m-1-2i}{k}\\times\\\\ \\binom{n(m-1-2(i+k))}{2j}\\binom{j}{t-n(i+k)} \\binom{n-1-s+t}{s-t}\\\\ =\\binom{mn-1-s}{s}. \\end{multline*} 2) The generalized Fibonacci polynomial $f_{m}(x,s)$ can be expressed as \\[ f_{m}(x,s)= \\sum_{k=0}^{\\lfloor (m-1)/2 \\rfloor}\\binom{m-k-1}{k}x^{m-2k-1}s^{k}. \\] We prove that the following functional equation holds: \\begin{equation*} f_{mn}(x,s)=f_{m}(x,s)\\times f_{n}\\left (\\,f_{m+1}(x,s)+sf_{m-1}(x,s), \\,-(-s)^{m}\\right) . \\end{equation*} 3) If an arithmetical function $f$ is multiplicative and for each prime $p$ there is a complex number $g(p)$ such that \\begin{equation*} f(p^{n+1}) = f(p)f(p^{n})- g(p)f(p^{n-1}), \\hspace{15pt} n \\geq 1, \\end{equation*} then $f$ is said to be \\emph{specially multiplicative}. We give another derivation of the following formula for a specially multiplicative function $f$ evaluated at a prime power: \\begin{equation*} f(p^{k})=\\sum_{j=0}^{\\lfloor k/2 \\rfloor}(-1)^{j} \\binom{k-j}{j}f(p)^{k-2j}g(p)^{j}. \\end{equation*} We also prove various other combinatorial identities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-28T21:51:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A55"
    ],
    "keywords": [
      "combinatorial identities deriving",
      "th power",
      "functional equation holds",
      "complex number",
      "prime power"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}